Modeling urban economics across a city's life cycle

00The thesis

Three things already exist in this repo, and polis is the missing middle:

mappa's placement answers where; it has no notion of a city growing an economy. A 4X needs cities with a legible economic history — a founding engine, a multiplier, a diversifying base, a place in a hierarchy. polis is that layer.

01The core economic engine — how cities start, and keep growing

This is the heart of polis, and the answer to how do cities start?

The big game in town — why a settlement nucleates

A city is not founded by spreading people evenly. It nucleates around one economic function strong enough to pull a surplus toward a point — and at the start, everything else is negligible against that one engine. The archetypes:

• Break-of-bulk / entrepôt. Cooley (1894): "population and wealth tend to collect wherever there is a break in transportation" — a head of navigation, a fall line, a lowest bridging point, a harbour, a pass, where cargo must transfer and be stored. The transfer is a captive choke point that extracts rent, and that rent is the founding engine.
• Staple extraction. A mine, a fishery, a fur/timber frontier — one commodity exported raw (Innis's staples thesis).
• Administrative / sacred seat. A temple or palace that concentrates tribute and redistributes it (the earliest Mesopotamian cities).
• The agricultural precondition (Childe, 1950): none of these can feed a town without a storable food surplus from the surrounding land. Surplus is the permission to urbanize; the export engine is the reason.

In every case the early city is one engine plus its support system — which is exactly why its earliest network is a star of routes to that engine (regime 1, below).

The first engine of growth — the economic-base multiplier

Economic-base theory (Sombart 1916 → Haig 1928 → Hoyt 1939; the export-base theory of regional growth, North, 1955) splits a city's economy in two:

• Basic (city-forming) — produces for export; earns income from outside. The big game in town.
• Non-basic (city-serving) — produces for the local market (the baker, the barber, the school). It exists because base workers spend their wages.

If s is the local-serving share, each export dollar re-circulates locally and total activity is the geometric sum:

M is the base multiplier — the structural analogue of the Keynesian income multiplier, with s playing the role of the marginal propensity to spend locally. If non-basic is half the economy (s = ½), M = 2: every export job supports one local job. If s = ⅔, M = 3. This multiplier is the core economic engine that allows for growth: add one base job, the town grows by M.

The second engine — why a city outgrows its founding base

The base multiplier is linear: double the staple, double the town. It cannot explain a city that keeps growing after its founding engine fades (the portage that silts up, the mine that empties). Two mechanisms make growth self-reinforcing:

• Agglomeration / increasing returns (Marshall, 1890): firms cluster for labour pooling, supplier sharing, knowledge spillovers (the "trinity"). Empirically doubling a city's size raises productivity ~3–8% (Rosenthal & Strange, 2004) — so size creates the conditions for more size. This is also New Economic Geography (Krugman, 1991, Nobel 2008): increasing returns + transport costs + mobile labour make a featureless plain bifurcate into a dense core and a thin periphery once transport costs fall past a threshold.
• Import-replacement (Jane Jacobs, The Economy of Cities, 1969): a city starts making what it used to import — which keeps that spending local and creates new local work, and the new producers soon export the new good. So the export base grows from within, endogenously, by "adding new work to old." Jacobs argued this — not the static export base — is the true engine of urban growth.

02The economy as a forcing on one field — three regimes

A city is always solving a flow problem on a graph. Which problem is set by the shape of its economy, and that shape changes over the life cycle. Both hoop solvers are the graph Laplacian L with different forcing — and the forcing tracks how concentrated the economic base is. It evolves one point → uniform → many peaks, so Physarum bookends the cycle (a star at birth, a network at maturity) with the diffusion/coverage solver in the middle.

03The life cycle

The phases are not invented — they track Wilbur Thompson's five stages of urban growth (1965): export specialization → the export complex → economic maturation (import-replacement) → regional metropolis → technical-professional virtuosity. polis reads that economic succession spatially, with the three regimes as its geometry.

why herePhase 0 · Site & the founding engine

Score every candidate cell on two independent axes (site vs situation): site = local terrain (water, defensibility, buildable land); situation = betweenness on the route/river graph. Some terrain features are forced spawn points because they manufacture a founding engine — the fall line (rapids = head of navigation and water power; one escarpment seeds a string of cities — Richmond, Baltimore, Washington), the head of navigation, the lowest bridging point (river port), the harbour, the defensive hill/meander/island, the gap/pass town. The founding advantage is a one-time tie-break, not a permanent subsidy — which is what later lets the city outlive the reason it was founded.

regime 1 · single attractor · PhysarumPhase 1 · Nucleus

The economy is a monoculture: one basic sector (the founding engine), the multiplier barely started. The network is therefore a star — sparse routes from the scattered hinterland to the one place that matters. This is the Physarum solver with current injected to a single sink and μ high (the field collapses to a minimal tree/star, not a grid). This is the regime you identified, and it is economically exact: a one-engine economy produces a one-hub network.

regime 2 · coverage · hypoxiaPhase 2 · Town

The base multiplier M = 1/(1−s) spins up the non-basic economy — bakers, smiths, a parish, a market. The binding constraint flips from route-to-the-engine to serve-everyone: every dwelling within reach of the well, the market, the mill, the gate. This is the angiogenesis solver (foam.js seize()): grow capillary lanes toward the worst-served ("hypoxic") cell until all tissue is within reach. Its grain follows Conzen (streets → burgage plots → buildings, each plot aging through the burgage cycle); the first land-use gradient appears via von Thünen (R = Y(p−c) − Y·F·m, uses sorted by the upper envelope).

regime 3 · demand · PhysarumPhase 3 · City

Agglomeration and import-replacement diversify the base into many sectors. The flow problem becomes one of moving concentrated flows among many origins and destinations — the Physarum solver again, with a dense, peaked O–D current matrix (gravity demand T_ij ∝ P_i·P_j/d_ij², Reilly 1931). Conductance adapts, dD/dt = f(|Q|) − D (Tero et al., Science 2010); arteries are the superlevel set; μ dials grid↔tree. Bid-rent (Alonso) sorts land use — concentric rings (Burgess), sectors (Hoyt), or multiple nuclei (Harris–Ullman). History freezes as fringe belts under a morphological-frame constraint. The loop closes (Levinson & Xie): road → accessibility → development → demand → more road.

the regionPhase 4 · Metropolis & the system

• Scaling (Bettencourt & West, 2007): Y = Y₀·N^β — infrastructure sublinear (β≈0.85), socioeconomic output superlinear (GDP, wages, patents, crime; β≈1.15), needs linear (β≈1.0). Pick N, derive the rest.
• Hierarchy (Christaller): threshold × range → hexagonal market areas; K=3 / 4 / 7.
• Size distribution (Zipf / Gibrat / Gabaix): proportionate growth + a reflecting floor → the rank-size law P_n ≈ P_1/n self-organizes.
• Diversification & resilience: the product cycle (Vernon) filters standardized work down the hierarchy; the portfolio effect of a diverse base buffers shocks — the structural basis of the size ratchet.

04The wider model — three clocks, two territories, two doors, two axes

The phases are the shape. The engine runs on three clocks, spans two territories, opens two doors to the wider world, and is judged on two axes. This is where polis becomes a hinterland sim as much as a city sim, and where it pulls in two more assets the repo already has.

Three clocks

• Continuous — growth on curves. Population and productivity don't step with the phases; they grow on an S-curve toward a carrying capacity K. The Verhulst logistic (1838), dP/dt = r·P·(1 − P/K), inflects at K/2; the Gompertz alternative dP/dt = r·P·ln(K/P) inflects early at K/e ≈ 0.37K — fast boom, long crawl to the ceiling. Northam's urbanization S-curve (1975) is the macro envelope.
• Exogenous — K is not fixed. Carrying capacity is "a function of how people live and the technology at their disposal" (Boserup; Haber-Bosch). So K(t) rises and the city chases a moving ceiling — stacked successive S-curves (the bi-logistic), not one plateau.
• Lumpy — infrastructure as projects. Capacity does not fill in smoothly. A bridge, wall, mill or grid is indivisible — a half-bridge carries nothing — so infrastructure arrives in lumps that step K up discretely. Hirschman (1958) splits investment into social-overhead capital (SOC) and directly productive activity (DPA): build SOC ahead of demand (permissive — invites DPA) or let it lag (compulsive — scarcity pulls SOC in). Rosenstein-Rodan's big push (1943): below a coordinated minimum quantum, returns are negative; cross it and external economies flip positive. Model K = K_continuous + Σ(infra_blocks), each block a binary paid out of surplus.

The tech tree — the master clock (and yes, we need one)

Technology is the one exogenous force that moves every other parameter: it raises the agricultural surplus ratio (freeing labour for the city), lowers transport cost (the master lever — as τ falls past a threshold it tips Krugman's core–periphery bifurcation: the dispersed equilibrium loses stability at the break point, agglomeration self-sustains past the sustain point), raises productivity, and cuts project costs. With general-purpose technologies (Bresnahan–Trajtenberg 1995), Schumpeter's creative destruction and Kondratiev's ~50-year long waves, it also rewrites which industries lead each era. Strip the tech tree out and the base multiplier just saturates against a fixed K — the city has no history. So a tech clock is structural.

Two territories — the hinterland coupling

A city and its hinterland are one coupled system — which is why polis is a hinterland sim as much as a city sim.

• The city zones the hinterland. von Thünen (1826): city demand, refracted through transport cost, sets the bid-rent gradient R = Y(p−c) − Y·F·m that sorts the land into concentric specialization rings (dairy → fuel → grain → ranching).
• The hinterland feeds the city. The hard ceiling: max city population ≈ (hinterland surplus + net food imports) ÷ per-capita need, the surplus itself raised by intensification (Boserup) and the agricultural tech chain.
• The catchment & the five forces. Christaller's complementary region; Jacobs' (1984) five forces a vital city radiates into its region — markets, jobs, transplants, technology, capital. All five → balanced city-region growth; only a subset → "stunted and bizarre." The city-region, not the city alone, is the unit.

Two doors — the broader economy (central place vs gateway)

Two distinct external growth drivers, through which raw resources, agriculture and away-market access reach the city:

So a polis city carries two size terms — a central-place term (local demand) and a gateway term (export-demand × break-of-bulk advantage). Raw-resource availability (where the staple sits on the mappa substrate), agricultural surplus (the hinterland), and away-market access (the gateway's reach) are the three inputs of the broader economy.

Two axes — scale and flourishing (the bloom and the dusk)

A city is judged on more than scale (population, GDP — the N^β laws). The second axis is flourishing, and it has two faces.

Livability (the floor). The civic-vitality model in hoop/econ — scoreSociety() → vitality 0–100 + tier on seven sub-signals (supply closure, multiplex thickness, social weave, Granovetter bridges, third-places, employment, resilience) — is the floor: is the city good to live in? Its framing already separates this from output: the loop closes, and "the real output is regard."

A structural edge (the bloom). But flourishing meant something sharper in Vienna 1900 (Schorske) — a cultural and technological efflorescence. Structurally that is the city's capacity to attract talent and generate technology, and it is the missing feedback that closes the growth loop: talent → human-capital externalities → innovation → import-replacement → new export base → scale → (more talent). Skilled people near skilled people produce more than the sum (Lucas: "what else is a city?"; Glaeser; Jacobs); Peter Hall's Cities in Civilization catalogues these golden ages; Florida names the levers Talent · Technology · Tolerance. And because technology diffuses, generating it first is a competitive edge — a lead-time that decays as it spreads. So flourishing is the city's tech-generation rate: it advances the tech clock for itself and leaks to laggards. Sassen's global cities are the limit case.

The dusk — the Owl of Minerva. Flourishing is non-monotonic, and its peak sits on the edge of collapse. Hegel: "the owl of Minerva spreads its wings only with the falling of the dusk" — Vienna's bloom was the eve of the empire's fall. The structural account is Tainter's declining marginal returns on complexity: a metropolis at peak complexity is at peak fragility, each added layer of coordination returning less until the margin goes negative (Thompson's ratchet has the same ceiling; Spengler's "megalopolis" the late form). So polis models flourishing with a complexity cost that eventually bites — bloom and dusk on one curve. (This resolves last pass's open question: flourishing does feed back into growth — via talent + tech — but the complexity cost can turn it negative at the top.)

05Financial flows — funding development at each phase

The economy above moves real goods and labour. Finance is the parallel system that moves claims on the future — it mobilizes surplus into investment, prices time and risk, and supplies the medium of exchange. It is not a garnish: it is how the lumpy projects get paid for and how the multiplier deepens, and its cost of capital is a master lever on par with transport cost — cheaper, deeper finance unlocks bigger projects sooner and is a decisive edge between cities.

What finance does — Levine's five functions

A financial system mobilizes savings · allocates capital · monitors its use · eases exchange · manages risk (Ross Levine). Schumpeter's slogan is the point: "the banker authorizes the entrepreneur" — credit creates the means for innovation before the returns exist (King & Levine, "Schumpeter Might Be Right," 1993). Financial deepening (Goldsmith's Financial Interrelations Ratio) tracks development; Patrick's supply-leading (built ahead of demand — the financial twin of Hirschman's permissive excess-SOC) vs demand-following finance names the two sequences. (Honest nuance: causation is contested — Robinson's "where enterprise leads finance follows," and the "too much finance" literature finds the effect turns negative at high depth — itself an Owl-of-Minerva point.)

Finance funds the three clocks

• Continuous clock: money + credit raise the velocity and reach of the base multiplier; the money multiplier 1/(1−reserve) is the structural twin of 1/(1−s).
• Lumpy clock — where finance bites hardest: a project builds when its NPV at the cost of capital ρ is positive and financing exists (tax flow + borrowing capacity). ρ is the gate.
• Who supplies the lump? Gerschenkron (1962): the more backward the economy, the larger the role of banks, then the state — scattered savers can't assemble the indivisible lump. The financial regime decides whether the big push is fundable at all.

The per-phase ladder

War, empire, tribute — the fiscal-military thread

Brewer's fiscal-military state (1989): war drove the mutually-reinforcing growth of taxation + public debt + bureaucracy. The polity that can borrow (capital-rich, credibly-committed) out-finances the one that can only tax or plunder — Tilly's capital↔coercion axis with a financial edge (Britain out-spent rivals while borrowing cheaper). Tribute is a forced periphery→core financial flow; conquest can repudiate the debt or seize the treasury — a financial shock.

The dark twin — financial crisis (the third discrete shock)

Finance has its own Owl of Minerva. Minsky: over a long prosperity the financing mix migrates hedge → speculative → Ponzi — "stability is destabilizing" — until a Minsky moment forces fire-sales. Kindleberger's anatomy fits 300 years: displacement → boom → euphoria → distress → panic/crash (Tulip 1637; Mississippi & South Sea 1720). The land cycle (Hoyt's ~18-year Chicago series) is the urban form. So finance adds a third discrete event-shock alongside conquest and plague — the financial crisis — and the flourishing financial centre is the most prone to it.

06The political register — defense, empire, conquest

Everything above models the city as an economic growth machine. The full sweep — the eXterminate of a 4X — needs the city as a political-military actor too. Most of it falls out of the economics already built; one piece does not.

Defense — a founding engine and a standing cost

Defense belongs in the early running, including nucleation. Site-selection already prizes defensive sites (hilltop, river-meander, island); the political register adds defense as a founding function — the citadel/acropolis or garrison that seeds a town (you cannot accumulate a surplus you cannot hold), on the same footing as break-of-bulk, the staple, and the administrative seat. The frame is Tilly's capital↔coercion spectrum (1990): polities form along an axis from the capital-intensive pole (cities, trade — the engine of everything above) to the coercion-intensive pole (war, conquest, empire) — "war made the state, and the state made war." A settlement carries a coercion parameter on that axis; defense is both a founding engine and an ongoing security cost paid from surplus.

Empire — an emergent configuration, not a separate engine

Is there an imperial version of the narrative, or can it be done by managing inputs and outputs? Mostly the latter. Empire is what the machinery already built does at high extraction-reach + coercion:

• An imperial capital is a super-gateway — the away-markets door run in reverse and at scale, extracting surplus from a vast periphery and returning less. This is Wallerstein's world-systems core / semi-periphery / periphery (1974), already encoded by the regional hierarchy + gateway terms; tribute is a forced, one-way reversal of trade.
• The primate city (Jefferson, 1939) is just a Zipf deviation — the top city pushed disproportionately large (q>1) by political concentration. A parameter on the size distribution, not new machinery.
• The failure mode is Kennedy's imperial overstretch (1987): military reach outruns the economic base — Tainter's negative marginal return in uniform.

So no separate "empire engine" is needed: empire is a configuration of the gateway, the hierarchy, path-dependence, and the coercion parameter.

Conquest — the one piece the clocks can't produce

Conquest is the exception: a discrete regime-change event, nothing the smooth growth curves, the lumpy projects, or the tech clock can generate. It is the city-scale cousin of hoop's removeImpact shock, with outcomes on a spectrum:

The outcome is size-dependent — exactly the intuition that a metropolis loses little. Locational inertia (the Bleakley–Lin logic: sunk capital + agglomeration persist through regime change) means a great city survives conquest and just changes masters — Rome, Constantinople, Damascus, conquered repeatedly, carry on. A mono-functional nucleus can die when its single engine is taken. Conquerors usually preserve rather than destroy — the city is the tax base — so urbicide is the exception. In model terms: conquest applies a shock whose damage is f(size, diversity, conqueror intent, whether the base sector survives) — a diversified metropolis absorbs it, a one-engine town may collapse to ruin.

07How the pieces compose into one engine

08Determinism, and the road to a 4X

Everything is seeded. (worldSeed, siteCell) → a city's entire economic history, identical on every machine, freezable to an ATProto record. No Date.now(), no unseeded Math.random() in generation. A city permalink (and, later, a 4X save) is only meaningful if the history reproduces.

polis is the economic map-and-history layer a 4X needs. Once a city has a legible economic history — a founding engine, a multiplier, a diversifying base, a place in a Zipf hierarchy — the 4X verbs act on something real: eXplore the mappa world, eXpand along the site-score field, eXploit the surplus and the export base, eXterminate over the central-place catchments. The phases are, deliberately, also the economic-era progression of such a game.

09Sources

Assembled from the canonical urban-economics, economic-geography, and complexity-science literature. Full reading list: